Exercise 33 Page 883

We are given the description of a game of tetherball and the formula for the relationship between the length of the string L and the angle that the string makes with the pole θ. L=gsecθ/w^2 We need to determine if the expression L= gtanθw^2sinθ is also an equation for the relationship between L and θ. First, notice that in both expressions there is g in the numerator and w^2 in the denominator. Let's isolate them to see which parts of expressions we need to compare. gsecθ/w^2 ⇔ g/w^2 * secθ gtanθ/w^2sinθ ⇔ g/w^2 * tanθ/sinθ Therefore, we just need to determine if secθ= tanθsinθ. To do so, we will transform the expression on the right-hand side. We will start with using one of the Quotient Ratios for tangent. tanθ=sinθ/cosθ Let's now use it to simplify the expression.

tanθ/sinθ

tan(θ)=sin(θ)/cos(θ)

sinθcosθ/sinθ

DivFracD

a/c/b= a/b* c

sinθ/sinθ * cosθ

ReduceFrac

a/b=.a /sinθ./.b /sinθ.

1/cosθ

Now, recall one of the Reciprocal Identities for secant. secθ=1/cosθ Therefore, after some transformations we have obtained that tanθsinθ=secθ. This tells us that the two given formulas are equivalent. gsecθ/w^2=gtanθ/w^2sinθ